INFLUENCE OF FINITE RADIAL GEOMETRY ON COHERENT
RADIATION GENERATION BY A RELATIVISTIC ELECTRON
BEAM IN A LONGITUDINAL MAGNETIC WIGGLER
Ronald C. Davidson
and
Yuan-Zhao Yin
PFC/JA-84-23
June, 1984
N
INFLUENCE OF FINITE RADIAL GEOMETRY ON COHERENT
RADIATION GENERATION BY A RELATIVISTIC ELECTRON
BEAM IN A LONGITUDINAL MAGNETIC WIGGLER
Ronald C. Davidsont
Plasma Research Institute
Science Applications Inc.
Boulder, CO, 80302
Yuan-Zhao Yin
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA, 02139
ABSTRACT
The influence of finite radial geometry on the longitudinal wiggler
free electron laser instability is investigated for TE mode perturbations
about a uniform density electron beam with radius Zb.
The equilibrium and
stability analysis is carried out for a thin, tenuous electron beam propagating down the axis of a multiple-mirror (undulator) magnetic field
It is
B [1+(6B/B )sink z],z, where X0 =21Tr/k 0 =const. is the wiggler wavelength.
assumed that k R<<1, and that perturbations are about the self-consistent
0
2m
Z-bmVb), where
Lb
L 2ybmb -2yb
Vlasov equilibrium fb ( , )nb
2_ 2 2
p =prU6' ,
is the canonical angular momentum, and nb
bW b'
Lb and Vb
are positive constants. For 6B/B 0 <<1 and slow beam rotation (wb
Wcb
eBO/y bmc),
Rb=(2T
the equilibrium density is uniform (%)
.
//Ybm2bb
are investigated
out to the beam radius
Detailed free electron laser stability properties
for the case where the amplifying radiation field has
TE-mode polarization with perturbed field components (6Ee,
B r, 6B z).
The
matrix dispersion equation is analyzed in the diagonal approximation, and it
is shown that the positioning of the beam radius (Rb) relative to the conducting wall radius (Rc)
can have a large influence on the growth rate and
detailed stability properties.
Analytic and numerical studies show that the
growth rate increases as Rb/Rc is increased.
t
Permanent Address: Plasma Fusion Center, Massachusetts Institute of
Technology, Cambridge, MA 02139.
* Permanent Address:
Institute of Electron, Academia Sinica, Beijing,
People's Republic of China.
2
I.
INTRODUCTION AND SUMMARY
There have been several theoretical 1-26 and experimental
27
-3 5
investigations of coherent radiation generation by free electron lasers
that use an intense relativistic electron beam as an energy source.
Both
longitudinal1-5 and transverse6-18 wiggler magnetic field geometries have
been considered, and there have been many theoretical estimates of the
gain (growth rate) that treat the system as infinite and uniform transverse to the beam propagation direction.
Few calculations,5,16,17
however, have attempted to include the important influence of finite
16 17
,
radial geometry in a fully self-consistent manner, and these analyses5,
have indicated that the relative positioning of electron
beam radius (Rb)
to conducting wall radius (Rc)
can indeed have a large effect on the linear
growth rate and the detailed stability properties.
The longitudinal magnetic wiggler configuration
1-4
has been identi-
fied1 as a strong candidate for coherent radiation generation at frequencies significantly higher than those generated by the cyclotron maser
instability (assuming the same average axial field B0 ).
In the present
article, we investigate the influence of finite radial geometry on the
longitudinal wiggler free electron laser instability.1-
The analysis is
carried out for a thin, tenuous electron beam propagating down the axis of
a multiple-mirror (undulator) magnetic field (Sec. II and Fig. 1).
It is
assumed that ko%<<l and that the amplifying radiation field has TE-mode
polarization with perturbed field components (6Ee, 5Br, 6Bz).
the beam radius,
Here, Rb is
\0= 27/k0 = const. is the wavelength of the wiggle in the
axial magnetic field,
and the conducting wall is located at radius r=R
.
A very important feature of the linear stability analysis (Secs. III and
IV) is that the positioning of the beam radius relative to the conducting
3
wall radius (as measured by k/Rc, say) can have a large influence on the
growth rate and detailed stability properties relative to the case where
the system is treated as infinite and uniform1 in the transverse direction.
To briefly summarize,- in Sec. II we describe the equilibrium
properties of a thin, tenuous electron beam propagating down the axis of
a multiple-mirror magnetic field.
Equilibrium self fields are neglected,
and the vacuum magnetic field is approximated by [Eq. (3)]l
B (r,z) = B 0
1 +
sinkz
,
B0(r,z) = 0
r
2-'2
for kR <<l.
Beam equilibrium properties ( /'t= 0) are investigated for
the choice of self-consistent equilibrium distribution function (Eq. (6)]36
nb
0
b
2A
5P2YblbPe
1
2YbmLb
2
2
2
where p 1 =pr+p8
:ebmVb)'
z is the axial momentum, P
is the canonical angular
momentum, and nb' Yb' wb' Tb and Vb are positive constants.
is the characteristic energy of a beam electron.
Here Ybmc
For small wiggler
amplitude (6B/B 0 <<l) and a slowly rotating electron
beam with Wb
eBO /Ybmc, it is found that fb
0
fb5
where V
2
T
2
TCh/Ybm.
LpJ
_Lb
abYb
vb
~
h%)
(1-r'/
~(Z
bbY b)
6(p
b '
In this case, the axial modulation of the beam
0
the beam density is uniform with nb()=
(9)].
tion (3)
Wcb
) can be approximated by [Eq. (16)]
envelope is negligibly small with Rb(z)b= const.
[Eq.
2
[Eqs. (10) and (14)], and
over the interval O<r<
The particle trajectories in the equilibrium field configura-
are also described in Sec.
II [Eqs.
(17)-(20)].
4
Stability properties are investigated in Secs. III and IV for TEmode perturbations about the choice of self-consistent solid-beam equiAssuming perturbed field components
librium specified in Eq. (16).
(5E 9 5B , 6B ), the linearized Vlasov-Maxwell equations [Eqs. (24)(25)] are analyzed using techniques established in recent investigations37-39 of the influence of finite radial geometry on the cyclotron
In particular, for a tenuous electron beam, we
maser instability.
approximate the r-depencence of 6E (r,z) by the vacuum waveguide solution, J 1 (t 0Zr/Rc), where a
is the Z'th zero of J1 (
0
Z) = 0 [Eq. (30)].
After considerable algebraic manipulation, this gives the matrix dispersion equation [Eq. (43)]
2
0
2
2-
-
It
+
c
Xn,n(k,w)
Rc
n=-
N#0
=F
n=-c
+
6E (k)
-N,n(k+NkOW)5 6 (k+Nk0
1-on
0,
,n(k,w) is defined in Eq. (50) for the choice
where the susceptibility
of radially confined equilibrium distribution fb
in Eq. (16).
For
sufficiently small wiggler amplitude, the off-diagonal terms in Eq. (43)
can be neglected, and the dispersion relation is approximated by (Eq. (47)]
2
- k2
a2
R
+
xnn (k,w) = 0.
c
The striking feature of the definition of \mn(kw) in Eq. (50) is the
strong dependence of the integrand on radial coordinate r.
For the case
of small energy variation over the beam cross section, we obtain the
approximate expression for ),(k,w)
given by [Eq. (55)].
5
2
(kw) =
4
2b
91_f_(
[
)
2
..b
]
1 - w-k~k )V-cb]
+
where f ()
(Wcb
g2f 2(w)
2 [-(kknnkOb-b
2)
c 2[w-(k+nk0 )Vb- cb]2
Wcb )
and f2 M) are defined in Eqs. (56) and (57), and
g
and g2
are the geometric factors defined in Eqs. (53) and (54) (see also Fig.
2).
In Sec. IV, we obtain approximate analytic and numerical solutions
to the diagonal dispersion relation (47) for Xn(kw) specified by
Eq. (55).
Several points are noteworthy from the stability analysis.
First, for a given harmonic number n, the characteristic maximum growth
rate [Eq. (68)] is largest whenever the argument of J
corresponds to the
first zero of J'(x)= 0, i.e., when
n
Wcb
kOVb
where a
6B
BO
n,l
is the first zero of J'(x) = 0. Second, the characteristic
n
1/3
maximum growth rate scales as (V /c)2/3^
nb
, which can lead to
n,l
substantial gain for the. longitudinal wiggler free electron laser..
Third, there is a strong dependence of stability properties on the
positioning of the beam radius (Rb) relative to the conducting wall
radius (Rc).
In particular, the growth rate Imw increases as Rb/R,
is increased, and the detailed dependence of Imw on
numerically in Sec. IV.
/R
is examined
Another important feature of the results is
6
that the instability is inherently broadband in the sense that many
harmonics are unstable, even when (wcb/k0 Vb)(6B/B ) is chosen to
maximize emission for a particular value of n.
Finally, also in Sec. IV, we investigate stability properties in
the limiting case where 6B=0.
This corresponds to the cyclotron
maser instability for perturbations about the solid electron beam
equilibrium f0( ,k) specified by Eq. (16).
The numerical results for
6B= 0 also show a strong dependence on radial geometry (Rbd/R)
7
II.
A.
EQUILIBRIUM MODEL AND ASSUMPTIONS
Equilibrium Field Configuration
In the present analysis, a tenuous electron beam propagates
along the axis of a multiple-mirror (undulator) magnetic field with
axial periodicity length X 0 = 27r/k
0
and axial and radial vacuum
magnetic fields, B (r,z) and B (r,z), given by1
r
z
B (r,z) = B
z
1 + 6BI
01
B0
kr)sink
0
z
ink0zj
(1)
B (r,z) = -6BI
(k r)cosk z
where I (k0 r) is the modified Bessel function of the first kind of
order n, and 6B/B 0 < 1 is related to the mirror ratio R by R
(1+6B/B 0 )/(1-6B/B 0 ).
=
In circumstances where the beam radius Rb is
sufficiently small that
k2
<< 1,
(2)
and the oscillatory field amplitude 6B is small with SB/B 0
<
1,
then the leading-order oscillation (wiggle) in the applied field is
0 =0
primarily in the axial direction, with B=
the context of Eq. (2),
O(k r6B/B )B
.
Within
the equilibrium magnetic field components
can be approximated by 1
B. (r,z) = Bo
+
B0
sinkoz,
j
(3)
B (r,z) = 0
r
8
-l
in the beam interior where r < Rb <<b k'
0
Assuming a tenuous electron beam with negligibly small equilibrium
self fields, then the electron motion in the longitudinal wiggler field
specified by Eq. (3) is characterized by four single-particle constants
of the motion.
These are:
pz '
2
2
2
'
r + pe
=
p.
2 4
2
ymc
P0
=
(m c
=
r p
2 2 1/2
2 2
+ c p1 + c P
-
)
(4)
,
A 0(r,z)
where pz is the axial momentum, p1 is the perpendicular momentum, ymc2
is the electron energy, and
0
1
A0(r,z) =-! rB
f1
+
B
sinkz
(5)
,
0
is the 6-component of vector potential consistent with Eq. (3).
Here,
-e is the electron charge, m is the electron rest mass, and c is the
speed of light in vacuo.
Note that ymc2 = const. can be constructed
2
from the constants of the motion pz and p,, which are independently
conserved.
It is important to keep in mind that the validity of the single22
< 1,
particle constants of the motion in Eq. (4) assumes that k0r
SB/B 0 <1, and that the oscillatory radial magnetic field B ~ -(8B/2)k rcoskz
0
To determine the range of
can be approximated by Br= 0 [Eq. (3)].
9
validity of this approximation, we have also calculated
(in an
iterative sense) the leading-order corrections to the longitudinal
0
and transverse orbits, treating the magnetic force (-e/c)vxB e
as
a small correction.
B.
Beam Equilibrium Properties
The TE-mode stability analysis in Sec. III is carried out for
perturbations about the self-consistent equilibrium distribution
0
(x
b
f
=
2
)5 (p
- 2bmT
- 22
Ybi.b6 (P
T 5 (P2
Ti
Ybmwbe
nb
A
where nb' T.Lb' 'b and Vb are constants, ybmc
2
bmV)
36
(6)
,
b b
const. is the
characteristic energy of a beam electron, and Vlb is defined by
Vib =(
2
b/ybm)
1/2
2
p
- 2ybMW Pe
.
Making use of Eq. (4),
2+
p-mr
p2r +
-yb
it is readily shown that
2
b2
(7)
+ y 2 m2 rwb
2
cb (1 +
-_B
sinkoz
-
b
where wcb = eBO/ybmc is the relativistic cyclotron frequency.
Eqs. (6) and (7),
From
it readily follows that the average axial and
azimuthal momentum of the beam electrons are given by
d3 p pf0
d
z b
<pz>
<p e>
rd33
YbmLvb'
fdd pp fbefb
fd 3pf0
d p
Moreover, it
f0 0
YbmWbr.
b
can be shown from Eqs. (6) and (7) that f
(x,
) corres-
10
36
ponds to a sharp-boundary equilibrium with density profile n0
T 0
d p fb given by
const.,
n (r,z)
0 < r < Rb(z)
(9)
=
r > Rb(z)
0
Here, the radial boundary Rb(z) of the electron beam is defined by
^2
2
Rb(z)
where
=2Tlb
=
-Lb
wcb [l+(6B/Bo)sink0 z]-
Wb
bm.
From Eq. (10),
0 < Wb
(10)
b
b is required to be in the range
cb[1-6B/B0(
for a radially confined equilibrium to exist.
Moreover, the maximum
occurs for k0 z=(2n+1)7r/2, n=±l, t2, ...
beam radius [Rb ]M
Introducing the effective thermal Larmor radius rLb
condition
k [Rb]2
22
0 Lb <
V -Lb
cb
the
<< 1 can readily be expressed as
b
Wb(1_L_
B0
cb
Wcb(
The stability analysis in Sec. III is carried out for the case
where the beam rotation is slow with
Wb
<<
(13)
W cb
and the wiggler amplitude is small with 6B/BO << 1.
In this case,
11
the axial modulation of the beam radius in Eq. (10) is very weak,
and Rb(z) can be approximated by the constant value
1/2
b (z)~Rb
(b
b cb )
(
Moreover, consistent with Eqs. (13) and (14), it is valid to approximate
Pe=-
which gives
b Mcbr 2/2 in Eq. (7),
P
2
2
-
Ybmwb P8-
2
YbmT'b
(15)
2
PI -
wcb and 6B << BO.
for wb <
Eq.
2
2 ^
(Ybmb) 2(1-r /2 )
The equilibrium distribution function in
(6) can then be expressed to the required accuracy as
0
b
nb
-
2
Tr
2
p2
(YbVb) (-r
b tb/%r
2
^ 2
(16)
bmvb
p-bm
2 2
where pi = pr+p2.
Equation (16) readily gives the rectangluar
density profile in Eq. (9) with constant beam radius equal to Rb.
Note from Eq. (16) that the equilibrium distribution function has an
inverted population 1 in pa with f0 = const.
x
2[p
- p2(r)]6(pY
mV
2 ^2
2
2
where plo(r) = (Yb mVlb) (1-r /Rb).
C.
Particle Trajectories in the Equilibrium Fields
The orbit equations for an electron moving in the axial magnetic
field B (z) = B +6Bsink z described by Eq. (3) are given by dp'/dt'=
z
0
-(e/c)v'B 0 (z'),
ymsv'(t')
0
dp'/dt' =
and y = (1+k'
2
x
(e/c)v'B0
2 2 1/2
/m c )
(z')
and dp'/dt'=0, where 2(t')
= const.
The axial trajectory
=
12
that passes through the phase space point (z,pz) at time t'=t
(z',p')
z
is given by
P
pz
,
z' = z + V
(17)
where T = t'-t and vz=p z/y m=const. is the axial velocity.
Defining
v' (t)=v'(t')+iv'(t') and making use of Eq. (17), it is straightforward
x
y
to show that v'(t') satisfies
V
d
=
dt'- +
C
~ c
1 +
B sin(koz+kov t)
'
),
z
O
0
(18)
where wc = eB 0 /ymc is the relativistic cylotron frequency for electron
Integrating Eq. (18) with respect to t'
motion in the average field B
and enforcing v'(t'=t)=v +iv =vlexp(i$), where (v ,v )
=
(vicosOv sin$)
1
is the perpendicular velocity at time t'=t, gives
'(t')
= vjexp (+iwT
(19)
.
c
SB cosk 0 z-cos(k 0 z+k0 v z)
B0
k vz
0
From Eq. (19), we note that
1v'(t')I
= const.,
the individual transverse velocity components,
as expected.
v' (t')
can be strongly modulated as a function of z and t'
wiggler field 6Bsink 0 z.
However,
and v'(t'),
by the longitudinal
Depending on the size of 6B/BO, this can
lead to a significant enhancement of radiation emission relative to
the case where
6B=0.
For future reference, it is convenient to Fourier decompose the
k0 z dependence in Eq. (19), making use of the Bessel function identity
13
exp(ib cosa)
JM(b)exp(-ima+ir /2)
=
where J (b) is the Bessel function of the first kind of order m.
This gives
viexp(i$)
v' (t'=
I
J
m,n
m
B
0 z B 0k
6B
B
n kv
n-
0
(20)
x
where
exp[i(WcT+mk0vzT)]exp[i(m-n)kOz]
.
mn denotes I' _
From Eq. (20), for (wc/k v)(6B/B)
of order unity, we note that the temporal modulation of the perpendicular
velocity can be strong at harmonics of k0 zk0Vb.
Finally, when Eq. (20)
is integrated with respect to t' to determine the transverse particle
orbit x'(t')+iy(t'),
we note that there are, resonant contributions
proportional to (wc+mk0 vz )
assumes that vz
(with w c+mkb0 b0)
.
n this regard, the present analysis
b is sufficiently far removed from cyclotron resonance
that the particle orbits do not exhibit large (secular)
transverse excursions.1
14
III.
A.
TE-MODE STABILITY PROPERTIES
Linearized Vlasov-Maxwell Equations
For the equilibrium configuration discussed in Sec. II, we now
make use of the linearized Vlasov-Maxwell equations to investigate
stability properties for electromagnetic perturbations with TE-mode
polarization.
It is assumed that a conducting wall is located at
radius r = Rc > Rb.
Moreover, perturbed quantities are expressed as
65(x,t) = 6$(x)exp(-iwt)
where Imc>O corresponds to instability.
symmetric
(3/H=O)
We consider azimuthally
TE-mode perturbations with electromagnetic field
components
5Ee(rz)~e
=
(21)
(
where (er'
e
=
Br (r,z),Ar + 6Bz(r,z),
) are unit vectors in the (r,e,z) directions.
Maxwell equations for 6
and 6
7x 6.,c = -
b
(22)
~
%c
d3 p v 6f
are given by
6 B
Vx6 E =
where 6J(x) = -e
=
I
o
The
41- e
3
d p v 6f
(23)
c
is the perturbed current density, and 6f (r,z,)
b
is the perturbed distribution function.
From Eqs.
(21) and (22),
5B (r,z)
15
and 6 Br (r,z) are related to SE (r,z) by
e
ri
6i z= -
(r6 E6
'
(24)
ic a
W az
r
6e
Moreover, substituting Eq. (22) into Eq. (23) and taking the O-component
gives
/2
2 r r
r
r
-E
+ 2
++
+(r,z)
-2
r
a
2
(25)
47riwe
3
d p v
i
6fb(r,z,,)
which relates 6E (r,z) to the perturbed azimuthal current density
6J (r,z) = -e
3
d p v
6b'
Making use of the method of chracteristics, the linearized Vlasov
equation for 6fb(r,z,,pt) = 6fb(r,zq)exp(-iwt) can be integrated to give
dt'exp [-iw (t'-t)
6 fb (r,z,,) = e
I
(26)
x
where (g',g')
v'x6B(x
6 (x ) +
c
*
b
,
)
,
'
are the particle trajectories in the equilibrium field
configuration (Sec. II.C) that pass through the phase space point
(g,,p) at time t'=t,
function.
and
(
) is the equilibrium distribution
We first simplify Eq.
(26) for the general class of self-
consistent Vlasov equilibria (Sec. II.B),
16
f0
0 (p-2y WP '
fb
be
Lb
b
(27)
),
where the perturbed electromagnetic fields have the components shown
in Eq.
After some straightforward algebra, the perturbed dis-
(21).
tribution function can be expressed as
3f 0 0
b
2e
6fb(r,zq)
0
-
where T=t'-t.
e
v'6iz(r',z')
dT exp(-iwt)
v e6Br(r',z')
(28)
0
fb
I
-
-
In obtaining Eq.
f 0/ap 2and af /ap
+1 v'6B (r',z')]
[E (r',z')
br'
br') +Yb
b
(p-
d- exp(-iot)
(28), use has been made of the fact that
are constant (independent of t')
along a particle
trajectory in the equilibrium field configuration. Moreover, from
2 2 21/2
=
V'=v
lm=p ad
are independent
(l+ /m c )
/ym and y' =y
Sec. II.B, v' = vz =
of t' in Eq. (28), and the axial orbit is given by z' = z + vz .
In addition, the transverse velocities v'
can be determined from Eq. (19),
v'(t'=t) =
= dr'/dt' and v'
a = r'd6'/dt'
subject to the boundary conditions
r, v (t'=t) = v,, r'(t'=t) = r and 6'(t'=t) = 6.
Equation (28) simplifies considerably in the case of very slow beam
rotation (wb «
Sec. II.B.
W cb) and weak wiggler field (6B << B0 ) discussed in
Neglecting the terms proportional to wb in Eq. (28), and
expressing 6Br in terms of 6Ee
0
af0
6fb(r,zq) = 2e
(24)], we obtain
[Eq.
d
-
exp(- i)p'6
r',z')
(29
0
0f
+
2p0
(29)
0
dTexp(-i)
p
I
6E(r',z),
17
where z' = z + v zT.
Eq.
(25)
In general,
Eq.
(29)
is to be substituted into
and the resulting equation solved as an eigenvalue equation
for SEa(r,z) and w.
B.
Matrix Dispersion Equation for a Tenuous Beam
For a low-density electron beam, we make use of the fact that
6
the solution for
Ee (r,z) is closely approximated by the vacuum
waveguide solution, and solve Eqs. (25) and (29) in an iterative
sense.
In particular, we express37-39
6t e(r,z) = 6E (z)J1 (aOr/RC)
where aOz is the Z'th zero of J1 (a)
6
0, and
E (r=Rc,z) = 0 at the
Here, Jn (x) is the Bessel function of the first kind
conducting wall.
of order n.
=
(30)
,
Taylor expanding the z'-dependence of 6E 9(r',z') in Eq. (29)
according to
6e (z')
=
k=-co
6E (k)exp(ikz+ikvt )
(31)
,
where k=2n/L and L is the periodicity length in the z-direction, we
readily find
0
3f0
/ kv
(k)exp(ikz)
d
6^f (r,zq) = e
b
k
8
2
z
l
W
bf+ k
9 2
ymw
0
b
pz
(32)
0
x
We substitute Eqs.
d-c
(30)
exp (-iWT+ikvz-p
and
(32)
into Eq.
(aOr '/Rc
(25),
make use of
18
{r
(3/3r)[r(a/ar))-r
}Ji (ar/R
0
oc dr r J (ao r/R )...
with
2
c
2
.
2 /R )J (ar/R ), and operate
This gives
2
Rc
az
(47re iW/c 2 ) 2 JO c dr r J
0
[RcJ 2 a 0o)/2]
dp
x
-~r)
(E
+
12(1 _ 2
V
(k)exp(ikz)
k
c
k
0z
x
r
dT exp(-iwr+ikvz
where use has been made of
J(a zr'/RC
Rc dr r J (a0 r/R)
/2)J
=(R
(a
We now evaluate the orbit integral
0I
dt exp(-iwT+ikvt )p I
The azimithal momentum p'
p=
(a0 r'/Rc)
is expressed as p,
is constant (independent of t')
p, = ymv
field configuration described in Sec. II.
(34)
= pjsin ('-e'),
where
for the equilibrium
Equation (34) then becomes
0
I
=
(35)
0 dT exp(-iWT+ikvzT)sin($'-e')Ji (a or'/Rc)
p
To simplify Eq. (35), we specialize to the case of small thermal
Larmor radius, r b/b << 1, and approximate the r' and e' orbits
by (r',6')
~
(r,e).
This is an excellent approximation for a
slowly rotating beam with wb/wcb
/wcb
<< 1 follows from Eq.
<
(14).
Eq. (35) can then be approximated by
,
inc
r2
^2
<< since rLb/R
1,
-2 /2
V b
^2
cbR
The orbit integral I in
19
I = pJ
1
f
(a0 r/RC)
dr exp(-iwT+ikvzr)
(36)
1 {expi($-e)]-exp[-i($'-e))
' is the rapidly oscillating velocity phase defined by [Eq. (19)],
where
'
'B
+=
cT
+ Wc
B0
+CoskO
)
- cos (k0 z+k~v
ko v
(37)
We substitute Eq. (37) into Eq. (36), make use of the Bessel function
identity in Eq. (20) to expand exp(±i$'), and then carry out the
integration over T for Imw > 0. After some straightforward algebra,
this gives
PiLJ1 ( 0r/R C)
2i
n,m
n
(
wc 6B
ovz 0)
B
kv
Jm
dB
B
zk E
n-m+l
exp~i($-8)]exp[-i(n-m)k0z](i)nM
X
(8
w-(k+mk 0 )vz~-
(38)
exp[-i($-e)]exp[-i(n-m)k0 z](i)-n+M+1
w-(k+mk0 )vz+Wc
where wc
=
eB0 /ymc, vz
=
pz/ym and
eigenvalue equation (33),
0
$,
d-
0
dpip
-"
L denotes
n,m 3n=-00
we express
d3
=
.
In the
m=-co
n
integration over
dpz and carry out the required
0
making use of ve=(p±/ym)sin($-e) and the fact that fb
0O 2
-
2 ^2
2
(ybmVib) (-r/
accuracy of Eq. (15).
is independent of $ to the level of
In particular, we readily obtain from Eq. (38)
20
2r
devisin($-
)I
-
1
4ymi
J
(a0 r/Rc)
JIrp
1
k
i
B exp[-i(n-rn)k Z]
0
cOv z B0 ) mk(0 vzB 0)
n,m n
(39)
n-m
w-(k+mk
w-(k+mk0 )v+W
z)~c
Substituting Eq. (39) into Eq.. (33), the eigenvalue equation
for 6E 6(z) can be expressed as
2
2
z
2
a
2
2
Rc
+2
e z
c
(40)
Xmn (kw)6E (k)
=k- m,n
x exp~i(k+mk0 -nk0 )z]
,
where the susceptibilityX mn (k,w) is defined by
=1
Xrn (k,w)
(4
2/c2
4[(R /2)J
(a0
r3p Jm (w
fd
\k Ov
[
09
dr r J2 (cL
0 r/R)
01
x
/n
B0 I
0d
'b
P a PP±
3 0
0
af
SV2[mm
BB)J
k-z0 BB
+ k
(
(i)nm
Iw-(k+mk0 )vz-W c
b
z
P±JJ
P.
-nf+
W-(k+mk 0)vZ +WC
(41)
21
In Eq. (41), vL = pl/ym, vz
=
PZ /ym, WC = eB0 /ymc and fd p = 2r f dpjpL
Equations (40) and (41) should be compared with Eqs. (22) and (23)
of Ref. 1.
The major difference is that Eqs. (40) and (41) include
finite radial beam geometry in a fully self-consistent manner, whereas
Ref. 1 assumes an infinite uniform beam in the transverse direction.
The effects of finite radial geometry are manifest through the occurrence
of the effective perpendicular wavenumber a
on the left-hand side
of Eq. (40), as well as the r-integration over J (a0 r/R ) and the
r-dependence of the equilibrium distribution function f in Eq. (41)
b
[see Eqs. (15) and (16)].
Eq.
(40)
Fourier decomposing the left-hand side of
with respect to z,
and changing the k-summation variable
on the right-hand side of Eq. (40) from k+mk0-nki>k gives
2
a2
--
k
-
e
k
(k)exp(ikz)
R2
c2
c
(42)
+
k m,n
Xm,n(k+nk0 -mk0 ,906
6 (k+nk0 -mk0 )exp(ikz)
=0.
Equation (42) gives the final dispersion equation
D (k,w)6
0
N#0
N=--
D0 (k'W
and
, and D
+
=
r
N#0
(k+Nko'
X1,+NO~)
(k+Nk0 )
(k+
0 ,
(43)
0
-l
where
e (k) ++
and XN are defined by
N=l
[2
-2
2
k2 ~a R
R(kw)
2J
+
c
(44)
dp.z
22
Xn-N,n (k+Nk0 ,W)
XN (k+NkO,)
(45)
.
The dispersion equation (43) can be used to investigate stability
properties for a wide range of system parameters.
To lowest-order,
the stability analysis in Sec. IV is based on a diagonal approximation
to Eq. (43), i.e.,
D0(k,w)
= 0
(46)
,
which neglects the coupling of 6 e(k) to the higher harmonic dependence
in 6E (k+Nk0 ) for N#0.
Making use of Eqs. (44) and (45), the
approximate dispersion relation (46) reduces to
2
k2
c 2R
a2
01
x
n=-
c
n,n
(k,w)
(47)
0
which is analyzed in Sec. IV.
C.
Susceptibility for a Nonuniform Beam
We now evaluate the susceptibility X m,n(k,w) defined in Eq. (41)
for the specific choice of radially nonuniform beam equilibrium in
Eq. (16).
The evaluation of x
(k,w) proceeds in a manner similar
to Ref. 1 with the important difference that the p,- and p Z-integrations
in Eq. (41) select spatially nonuniform values of p, and ymc
choice of f
in Eq.
(16).
for the
In this regard, it is convenient to define
23
PLb = Ybm^Vib
Pzb
=
Yb=
YbmVb
(48)
'
/2
21-
222
(1+p2b/M2c2)1/2
b;/
zb
-1/2
2
(Heretofore, yb has been a convenient scale factor, with Ybmc
measure to the characteristic electron energy.)
Eq. (16) that the delta-function form of f
b
a
It is clear from
selects the values of
perpendicular momentum p, = plO(r),
total energy ymc2
perpendicular velocity v, = p /ym
1
V10(r), and axial velocity vz
p /Ym
=
=
=
Y
r)mc2
Vbo(r), where
2
2
r2
^2 b
V.LO(r)
Lb
b 1 +
YO~r
-
c
1/2
r
(49)
Rb1/2
RLb(b-r
Y(r)
Yb
Vb (r) E Vb
bO
b y0 (r)
for 0 < r < Rb.
Note from Eq. (49) that p1 0 (r) and V1 0 (r) are highly
nonuniform across the beam cross-section, assuming maximum values
on the axis of the beam (r=O), and minimum values of zero at the edge
of the beam (r=
).
(r=b).
theothr hand, for Vib/c 2 << 1, the variation
On the other hndfor2
of y 0 (r) and VbO (r) is relatively weak over the beam cross section.
Substituting Eq. (16) into Eq. (41) and carrying out the
integrations over p,
and pz,
we find after some straightforward but
24
tedious algebra that Xm
(k,w) can be expressed as
2
- 4
Xm,n(kw)
c
0
c
0
bc
2
Rb dr r J2 (
0
[(R /2)J2 (a0
2
2)
O(r)
(k+nk 0-mk )c
0+
YO
Vb
b2
c
V2
0
(50)
]
-
V2
cb~r)b
c2
+ (cb
where
2
pb
be
in Eq. (49).
2
w-(k+nk 0)V yOwcb
'-W cb)
b,
(w0-(k+nk -mk )(k+nk )c 2i
JI
0
yb/YO
~
wcb = eBO/Y bmc, and YO(r) and VIO(r) are defined
Here, emn is defined by1
n
E
m,n
e6B \
(k=Z)
zp
m/e6B
kcpz
z)
z)z
(51)
b
The r-dependence of the integrand in Eq. (50) is generally quite complicated.
For present purposes, we assume Vj,/c <<1 and approximate
YO(r) = Yb in Eq.
(50).
This gives the approximate result
25
Xm,n (k,w)
m
I
(
(
=2
2
4C
[(R /2)J
cb
k0 b B
+
2[w-(k+nk -mk
0
(1
(52)
m,n
)
2-(k+nkmk 0)(+nk)c2
w-(k+nk 0 )vb
cb
r2
^2
+ (Wcb -b
(52),
L)
2
C
In Eq.
ao9
b
(k+nkO-mk)c
b
r2
VLb
2
2
dr r J
0
6B
kOVb B 0
n
n-m
(a0 )]
( cb
B
w-(k+nk0 )-wcb
b
lb
-
1
'wc)
the r-dependence of the integrand is of the form
We'introduce the geometric
rJ1 (aor/Rc) and r2 (a2r/Rc)(1-r/)
factors g, and g 2 defined by
1A
g2
J (aOr
dr r
2
[(R /2)J (a O)]
0
R
(53)
2
+
(a0
cJ (OR
(1
2
J
a0%
R)
and
1
2
2 [(R /
2
2
[dr r
2
J0
a%]
f
2 2
3R 2 2 (
R
c 0
+~
0
a0%
2
1
a
OR Rc
)(
1
kb
09 R
c
1
OR R
2R
C
(54)
26
Note
from Eq.
the
that g 1 =1 for a uniform density beam filling
(53)
waveguide (Rb/Rc=l).
On the other hand, for Rb=Rc, it follows from
Eq. (54) that g2 = (2/3)
Figure 2 shows plots of the
.
geometric factors g1 and g2 versus ^b/Rc for several values of Z.
Note that g, and g 2 increase monotonically for increasing Rb/Rc.
Substituting Eqs. (53) and (54) into Eq. (52), we can simplify
the expression for Xmn (k,w) for general values of m and n.
In the
approximate dispersion relation (47), however, only the diagonal terms
(52) -
For m-n, Eqs.
are retained.
(54) give for the
susceptibility
2
Xn,n (k,w)
=
Mcb
k b
-bZ2
4 2
n
c gb
6B
B
0
w-(k+nk 0
b
4brfi e
=
2
cb
b
g
[-(k+nk 0 Vb-Wb 2)
(55)
wcb
+ (cb
Here, w2b
gf(W)
2
glf 1 (W)
x
S
,
b
=
and
are defined in Eqs.
(53) and (54), and fI(w) and f2 (w) are defined by (for m=n)
f (W)
=
2
Sc
(w-kVb) +
^22
Vitb
g2 kc2
Lb 92
21
f2 M) = w2 - k(k+nk 0 )c2
,
b
(56)
n,n
.57)
In overall form, Eq. (56) is similar to the result obtained in Ref. 1
for the case of a uniform beam with infinite cross section.
There are
important differences, however, associated with the dependence of
Xn,n (k,w) on the geometric factors g, and g 2 '
27
IV.
ANALYSIS OF DISPERSION RELATION
A.
Approximate Dispersion Relation
Making use of the expression for
Xn,n (k,w)
in Eq.
the
(55),
approximate dispersion relation (47) for a tenuous beam can be expressed as
2
-2
2
0
2-
k2 _
k
(,
/
2
p
c2
c
=
n=--C
cb _B
2
nnwkOb
k V B0
2
V1f(W)
ib.
c2
[1w- (k+nkO)b
[2
(2(W)V
[W-(k+nk 0 )Vb
+ (cb
2
)
(58)
cb]2
Wcb)
The dispersion relation (58) clearly has a very rich harmonic content,
and Eq. (58) can be solved numerically retaining several terms in the
For present purposes, we assume that the harmonics in
summation over n.
Eq. (58) are well isolated, and investigate stability behavior
near cyclotron resonance with polarity
w -
(k+nk O)Vb = +wcb
(59)
'
for a particular choice of harmonic number n.
Equation (58) can then
be approximated by
c2/R )[w-(k+nk
(w2 -c2 k-a
=1 2
4
k
b 2
cb]
6_B-
2 ("cb
pb' n
bV
0
nk)b
Ok c
b
(60)
)
v2-
x
Vib
gf1(W) [W-(k+nk O)Vb-wb] - 2
c
(9
92 f 2 (W
28
where g,
g2
fl, and f2 are defined in Eqs. (53),
(54),
(56), and (57).
For w = (k+nk)Vb+ cb' the first term on the right-hand side of
Eq.
(60) is negligibly small, and the dispersion relation can be
approximated by
(W2c
2
2
2
k2 c2 k )
[w-(k+nkO
b-b]
1 2
1b 12 Icb6B(1
+ 4 g 2 w b~
\ObB)
(61)
^2
1
2
Vib
4 52"wpb
The dispersion relation
2
2 (wb
~n
B
SOb BO
(knkc
2
nc
-k2c2
Z
(60) is solved numerically in Sec. IV.C,
and analytic estimates of the instability growth rate are made in
Sec. IV.B.
B.
Analytic Results
For a tenuous beam with (w2b/c k )(2b/c2) <<
1, we look for
solutions to Eq. (61) near the simultaneous zeroes of
S=
(c2 k +c2 kZ)1/2
(62)
w = (k+nk0 Vb + W cb
Here,
k ±Z= a Z/Rc is the effective perpendicular wavenumber, and we
have chosen the branch with positive frequency
(w>O) in Eq.
(62).
Shown in Fig. 3 are plots of w/ck 0 versus k/k0 obtained from Eq.
for the choice of parameters kZ/k0 =9.58, Vb /c=0.86
and several values of the harmonic number n.
6
, and
(62)
cb /cko=4.783,
Denoting the simultaneous
29
,ik ),
solutions to Eq. (62) by (
we find for the upshifted frequency
and wavenumber
2
2
=Y2 (n
(i
W+a
b2n
0 b cb
1/2
J
k Z
2
Yb(nkOVbcb
1
+b
1(3
,
(63)
and
1/2
2 2
k
2 =b(nk
b
bbWb+
1 -
+k
Yb
where 8
b and y b are defined by
Z
_____k
k0 b
(64)
cb
b/c
anddb
b
b
J
_
(-$2)-1/2.
2
(1% bot
Note
that (n ,k ) corresponds to the uppermost intersection points in Fig. 3.
Moreover, for k± -O, Eqs. (63) and (64) reduce to the familiar results
obtained in Ref. 1 for a uniform density beam with infinite transverse
dimension (Rb,Rco).
For solutions to exist, it is clear from Eqs.
(63)
and (64) that the inequality
22
c K
=
2 2
c a0
2
2
yb(nk OVbwcb)
<
2
,
(65)
c
must be satisfied, which we assume to be the case in the remainder
of Sec. IV.
For purposes of making an analytic estimate of the growth rate,
we express
w=! +6w and k=k +6k in Eq. (61),
weexrsn
as a small parameter.
W- C2
c2
treating (w2
(lb/
n
6k)
k
2)(2
2
n (VLb/c)
To leading order, this gives
(6w-Vb6k)
2
2 VLb
1
8 '22 pb c2
+
1
I
gw
g2wpb V
2
2 (knnk0-k.L )c
n (k +k
nI
)1/2
L
2
(66)
30
denotes J2 (cb6B/k
where
[Eq. (62)], and (o
VbBo), use has been made of(! = (c k+c2 k
in Eqs. (63) and (64).
n ,kn ) are defined
1)
As a simple
analytic estimate of the characteristic maximum growth rate, we consider
Eq.
Typically, the Jn contribution on the left-hand
(66) for 6k=0.
side of Eq. (66) makes a small contribution in comparison with the
right-hand side, and Eq. (66) can be approximated by (for Sk=0),
r3
n
3
2
1
8
2
Lb
2
2wpb
2
c
-k)
2 (knnk 0
1/2
2
2
n
(67)
(kn+kig)
c
2
For sufficiently short emission wavelength that knnk0 > kI,
Eq. (67) gives
V3
I2~ =-
V3
n ~4
Re6w
=-
2
22
2 Lb (knnk0-kZ)c
2
22wpb n 2
2 "n
c2
^ 2 2 1/2
(k +k 2 9)1 n
1/3
'
(8
(68)
2n
Several points are noteworthy from Eq.
(68).
First, for a given
harmonic number n, the characteristic maximum growth rate Im6w defined
in Eq. (68) will be largest whenever the argument of Jn corresponds
= 0, i.e., when
to the first zero of J'(x)
n
Wcb
6B =
k0 b B0
where a
n,l
as
(69)
n,l'
(x)=0.
is the first zero of J'
n
(\±b/c)2/31/3,
can be substantial.
Second, since Im6w scales
the growth rate for the longitudinal wiggler FEL
Third, the growth rate increases as the geometric
factor g2 is increased (see also Fig. 2).
electron laser instability described by Eq.
Moreover, the free
(61)
is inherently
31
broadband in the sense that many harmonics are unstable, even when
(Wcb/k0Vb) (SB/B 0 ) is chosen to maximize emission for a particular
value of n [Eq.
wavelength with
(69)].
Finally, in the limit of very short emission
2
kik, it follows from Eq. (68) that Imdw can
nnk 0 >
be approximated by
1/3
Im6 W =
(g 2 wbnkocJn
C.
.Lb
(70)
Numerical Results
The dispersion relation (60) is a fourth-order algebraic equation
for the complex eigenfrequency w.
Equation (60) has been solved
numerically over a wide range of system parameters.
In this section,
we summarize selected numerical results for 6B/B0 # 0 (longitudinal
wiggler free electron laser) and for SB= 0 (cyclotron maser).
Longitudinal Wiggler Free Electron Laser:
Shown in Figs. 4 and 5
are plots of the normalized growth rate Imw/ck 0 versus k/k0 obtained
numerically from Eq. (60).
AbOl
are kO
Yb
2
, aOi =3. 8 3 (Z=1), ViJb/c =0.1,
Spb/ck 0=0.25, and 6B/B0
(6B/BO (cb/ck0) = a 1
The parameters common to both Figs. 4 and 5
=
1/3.
Vb/c= 0.8 6 6 ,
In Fig. 4, we have chosen the parameter
= 1.841, which maximizes the growth rate for n= 1.
0
On the other hand, in Fig. 5 we have chosen (6B/Bc
which maximizes the growth rate for n= 3.
cb/ck0 )=a3,1=
4
.2 01,
For example, comparing Figs.
4(a) and 5(a), we note that there is a considerable upshift in maximum
growth from normalized wavenumber k/k0= 31.5 in Fig. 4(a) (n=1) to
k/kO
97.3 in Fig.
5(a)
(n= 3),
from 1.841 to 4.201.
as the parameter
(6B/B(
cb/ck 0)
is increased
Figures 4 and 5 also illustrate the strong depen-
dence of growth rate on finite radial geometry.
In particular, the plots
32
in Figs. 4(a) and 5(a) are presented for the case where the conducting wall
is relatively close to the electron beam (Rb/Rc=0.25, gl=3.792 xl0 2
and g 2 =1.314 x10-2 ), whereas the plots in Figs. 4(b) and 5(b) correspond
to the case where the conducting wall is much further removed from the
electron beam (Rb/Rc
-4
-3
and g2 = 3.704 x 10).
0.1, gl 1.105 x 10,
As
/Rc is decreased, we note that there is both an upshift in the value of
k/k0 corresponding to maximum growth as well as a significant reduction in
[Compare Figs. 4(a) and 4(b) or Figs. 5(a) and 5(b).]
growth rate.
It is
clear from the analytic results in Secs. IV.A and IV.B and the numerical
results in Figs. 4 and 5, that there is a strong influence of radial
geometry on the growth properties of the longitudinal wiggler free electron laser.
Indeed, as expected from Figs. 2 and Eqs. (60) and (61),
the
closer the conducting wall (Rc) is positioned to the beam radius (s),
larger the instability growth rate becomes.
the
Of course, this is associated
with the fact that g2 increases as Rb/Rc is'increased [Fig. 2 and Eq. (54)].
Another striking feature of Figs. 4 and 5 is that the instability is
relatively broadband.
Even though (6B/B )(Wcb /ck0 ) is chosen to maximize
Imw for a particular value of n, adjacent harmonics still have relatively
strong growth.
(See also the discussion at the end of Sec. IV.B).
Cyclotron Maser Instability:
In the limiting case where 6B= 0 and the
applied magnetic field is given by the uniform value B9z,
term survives in Eq. (60) with J (0) = 1.
only the n= 0
The resulting dispersion rela-
tion applies to the cyclotron maser instability for the choice of solid
electron beam equilibrium distribution in Eqs.
(6) and (16).
The dispersion
relation (60) has been solved numerically for the growth rate Imw assuming
6B= 0.
Typical results are summarized in Fig.
6,
where the normalized
growth rate Imw/wcb is plotted versus kRb for the choice of parameters
33
Yb = 2,
a % =3.83 (Z= 1),
Vb /c0.1, Vb/c =0.866 and Wpb Wcb = 2.29 x 10-2
To illustrate the influence of finite radial geometry, we have chosen
Rb/Rc = 0. 2 5 in Fig.
1.314x 10-2
and
-2
6(a)
(corresponding to g, = 3. 792 x 10-
and g2
b/R= 0.1 in Fig. 6(b) (corresponding to gl= 1.105 x
10-3 and g2 = 3.704 x 104 ).
free electron laser (B#
0),
As for the case of the longitudinal wiggler
it is clear from Fig. 6 that the relative
positioning of the beam radius (Rb) and the conducting wall radius (Rc)
also has a large influence on the strength of the cyclotron maser
instability.
[Compare Figs.
Indeed, the growth rate increases as {/Rc is increased.
6(a) and 6(b).]
As in Figs. 4 and 5, there is a con-
commitant upshift in k/k0 corresponding to maximum growth
as Rb/Rc is
decreased.
34
V.
CONCLUSIONS
In this paper, we have investigated the influence of finite radial
geometry on the longitudinal wiggler free electron laser instability.1
The analysis is carried out for a thin, tenuous electron beam propagating down the axis of a multiple-mirror (undulator) magnetic field (Sec.
It is assumed that k2
III).
<<l and that the amplifying radiation
field has TE-mode polarization with perturbed field components (6E,'
6B ).
6B,
A very important feature of the linear stability analysis
(Secs. III and IV) is that the positioning of the beam radius relative
to the conducting wall radius (as measured by Rb/Rc, say) can have a
large influence on the growth rate and detailed stability properties
'
for perturbations about the choice of equilibrium distribution f
in Eq. (16).
Several points are noteworthy from the stability analysis in Sec.
First, for a given harmonic number n, the characteristic maximum
IV.
growth rate [Eq. (68)] is largest whenever the argument of J
sponds to the first zero of J'(x)= 0, i.e., when (wcb/k
a
n,l'
where cc
n,l
rate scales as
corre-
Vb)( 6 B/BO
is the first zero of J' (x) =0. Second, the growth
n
2/3^ 1/3
, which can lead to substantial gain for
nb
(VJLb/c)
the longitidunal wiggler free electron laser.
Third, there is a strong
dependence of stability properties on the positioning of the beam
radius (R)
relative to the conducting wall radius (Rc).
In particular,
the growth rate Imw increases as Rb/Rc is increased, and the detailed
dependence of Imw on Rb/Rc was examined numerically in Sec. IV.
Another
important feature of the results is that the instability is inherently
broadband in the sense that many harmonics are unstable, even when
(Wcb/k0 b )(6B/B
of n.
0
) is chosen to maximize emission for a particular value
35
Finally, also in Sec. IV, we investigated stability properties in
the limiting case where 6B= 0.
This corresponds the cyclotron maser
instability for perturbations about the solid electron beam equilibrium
f0(x,)
specified by Eq. (16).
The numerical results for 6B= 0 also
exhibited a strong dependence on radial geometry (R./Rc).
ACKNOWLEDGMENTS
This research was supported by the Office of Naval Research.
36
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1.
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38
FIGURE CAPTIONS
Fig. 1
Longitudinal wiggler free electron laser configuration and
coordinate system.
Fig. 2
Plot of geometric factors (a) g1 [Eq. (53)] and (b) g 2 (Eq. (54)]
versus R./Rc for several values of Z.
Fig. 3
Plot of w/ck0 versus k/k0 obtained from Eq. (62) for yb
Vb/c =0.866, aO= 3.83, wcb/ck= 4.78 and k Rc = 0.4.
interception points (wn
k
Fig. 4
=
2,
The upshifted
n ) correspond to Eqs. (63) and (64), where
a k/RC
Plot of normalized growth rate Imw/ck0 versus k/k0 obtained
b
numerically from Eq. (60) for kORb0.,
Vb /c=0.866,
(6B/B0
2,
/c = 0.1,
at = 3.83 (Z= 1) wpb /ck = 0.25, 6B/B = 1/3, and
cb /k0 V) = 1.841 for (a) k0R
c = 0.
25,
g,=
- 20c
3.792 x 10 -2 and g 2 = 1.314 x 10-,
and for (b) k R
= 1.0,
R/Rc
-4
0.1, g=1.105 x 10 -3 and g 2 = 3.704 x 10
Fig. 5
Plot of normalized growth rate Imw/ck0 versus k/k
'1'
numerically from Eq. (60) for kO
Vb/c= 0.866, a
0 P,
x
pb
10 2 and g2= 1.314 x 10- 2,
0.1, g, = 1.105
/ck
x
10
3
b = 2, V b/c = 0.1,
=0.25, 5B/BO = 1/3,
00
and
Rb/Rc= 0.25,
=
and for (b) k Rc = 1.0,
/R
4.201 for (a) k Rc= 0.4,
(6B/BO)(b/ck)
3.792
=3.83 (Z= 1) , a)
obtained
and g2 = 3.704 x 10
4
39
Fig. 6
Plot of normalized cyclotron maser growth rate Im/w cb versus k
obtained numerically from Eq. (60) for 6B= 0, yb
2,
V Lb/c = 0.1,
V/c =0.866, a =3.83 (Z= 1), and wpb cbb = 2.29 x 10-2 for
09.
b
(a) Rb/R=O. 2 5 , g 1 372l9
-2= and for
g 2 =3.314 x 10-,
=and
(b) Rb/Rc
.1,
1.105 x 10 3 and g 2 =3.704x10.
40
I
I
I
I
I
I
I
I
,%O
0
-
I-
i
I
lob
L
i
*-~L)
z
0
cr-1
uw
-j
wi
Fig. 1
~mJ
41
1.0
0.8-
0.6-
g1
0.4-
0.2-
1=4
.
=2
1=1
f=3
0
0.2
0.4
0.6
A
R b /R c
Fig.
2 (a)
0.8
1.0
42
0.8
0.6-
9 2 0.4-
0.21=4
f=l
t=2
1=3
0
0.2
0.6
0.4
Rb/Rc
Fig.
2 (b)
0.8
1.0
43
I
60-
Yb= 2
ao
50-
I
1
Vb
-:- =0.866
=c4.78
= 3.83
koRc = 0.4
40-
ckO
30-
n=4
n=3
n=2
20-
10-
0
0
0
30
k /ko
20
Fig.
3
40
50
60
44
IfI
in
In
'~J.
ro
I,
-
0
0
to
N~
6
W)
II
C
0
a
11
-In(
<
col
0t
0
0n
M
II
C
0
if
if)
N~
IT
I
I
In
I
Vp
.0-01/('WT
I
to
101
I
Nc
I
0
45
(0
LO
C~j
0I
04/I
0
rI
I
46
0f
II
0
C
0
0
I,
C
Lc)
0
cli
0
N
6
S
<>0
ro
co
it$
C;
11
II
a to
Jol
a)
-
C
m7
0
>(0
i
N
0
0
0
I
C0
I
I
I
to)
N
OJlM
Irr0
I
0
47
I',
0
CC
00
CC
.0-
OO/mw
2
0
48
7
Vb
--
Yb =
61-
=0.866
cC
V.b= 0.1, aot
Wpb
8B=0,
= 3.83
=2.29x-2
Wcb
5k
(a)
Rb
=025
4k
3
3
E
0
3k
2k
I F-
0
.-V
I.'
Fig 6(a)
I
7.0
I
7.5
kR b
I
8.0
8.5
49
(b)
1.5
Rb
-c
4'
.0
0
3 1.0
N
3
E
q~.
0
0.5
.--- j
01
I I
I,
8.0
7.5
A
k
Fig.
6(b)
Rb
|
8.5
9.0